Beamforming devices and methods

ABSTRACT

Devices and methods are provided for directionally receiving and/or transmitting acoustic waves and/or radio waves for use in applications such as wireless communications systems and/or radar. High directional gain and spatial selectivity are achieved while employing an array of receiving antennas that is small as measured in units of the wavelength of radio waves being received or transmitted, especially in the case of spatially oversampled arrays. Frequency/wavenumber, multi-dimensional spectrum analysis, as well as one-dimensional frequency spectrum analysis can be performed.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of a prior U.S. patent applicationSer. No. 14/587,160, filed Dec. 31, 2014, which is a continuation ofprior U.S. patent application Ser. No. 14/053,512, filed Oct. 14, 2013,now U.S. Pat. No. 9,154,214, which is a continuation of prior U.S.patent application Ser. No. 13/301,393, now U.S. Pat. No. 8,559,456,which is a continuation of prior U.S. patent application Ser. No.12/372,454, filed on Feb. 17, 2009, now U.S. Pat. No. 8,064,408, whichclaims the benefit under 35 U.S.C. §119(e) of U.S. Provisional PatentApplication No. 61/030,013, entitled Beamforming Architecture, filedFeb. 20, 2008; 61/058,999, entitled Application of Harvey Matrix toReplica Correlation/Matched Filtering, filed Jun. 5, 2008; and61/122,503, entitled Beamforming Device and Method using HermeticTransform Processing, filed Dec. 15, 2008, all of which are incorporatedherein by reference in their entirety.

BACKGROUND

The present disclosures relate to a device and method for directionallyreceiving and/or transmitting radio waves.

Signals from a plurality of receiving elements configured as an arraycan be delayed, or equivalently phase-shifted, and combined (summed) tofilter out signal arrivals from particular directions in order to createa set of directional ‘beams’ in physical space. Such beams may be usedto improve the reception of signals in an interference background, suchas the reception of cellular telephone signals in a radio frequencybackground dominated by signals originating within the same spatial cellof the mobile telephone network or from adjacent cells.

The same principle works for transmission of direction beams of radiowaves as well. In this case, a signal to be directionally transmitted isreplicated and phase shifted or delayed, as well as amplitude weightedand fed to the individual transmitting antenna elements, in order toproduce desired directional beam characteristics.

For a uniformly sampled array, a fast and efficient form of digitalprocessing that is commonly used to perform beamforming is thewell-known ‘Fast-Fourier Transform’ or FFT, a fast version of theDiscrete Fourier Transform, or DFT, and known as the ‘Butler Matrix’ inantenna theory.

With DFT/FFT processing, the degree to which frequencies or spatialdirections can be resolved, known as the ‘resolving power’ of thesystem, is normally understood to be limited by the amount of signalobservation time (the number of samples times the sampling timeinterval) in the case of frequency spectrum analysis; or in the case ofbeam formation, by the aperture size/array dimensions. For example, inthe case of a linearly arranged set of receiving elements, an angularbeam width (i.e., the spatial resolution for resolving signals arrivingfrom differing directions) for a sufficiently well sampled aperture(spatial sampling no greater than one-half wavelength) is given by thewell-known ‘diffraction limit’, θ=λ/L, where θ is the beam width, λ isthe wavelength corresponding to the incoming signal, and L is theaperture (array) dimension. This limitation often presents a problem tothe system designer to whom this limit may appear to be a hard and fastlimitation on either system size for a fixed beam resolution requirementor on system resolution where system size is constrained.

In the conventional process of beam formation, weighting factors areoften applied to the channel data prior to the DFT transformation forthe purposes of reducing spatial response in directions far from themain lobe of the spatial beam pattern. Beam response is reduced onsecondary side lobes of the pattern, while the main lobe of the patternis somewhat widened. See Monzingo and Miller, “Adaptive Arrays”, JohnWiley & Sons (1980), p. 274 (Butler Matrix).

In so-called ‘super-gain’ or ‘super-directive’ systems, weightingfactors are applied to the channels, prior to the DFT transformation,for the purposes of narrowing the main lobe of the beam, especially inthe case where the spacing between receiver elements is less than therequired half wavelength. Weighting factors can be chosen in such casesto produce a consequent reduction in beam width (i.e., an increase inresolution), but in general with a loss of system sensitivity andbandwidth. See Doblinger, “Beamforming with Optimized InterpolatedMicrophone Arrays”, IEEE HSCMA Conference Proceedings (2008), pp 33-37.

SUMMARY

The present disclosure relates to devices and methods for directionallyreceiving and/or transmitting radio waves for use in applications suchas wireless communications systems and/or radar. In particular, devicesand methods are provided for achieving higher directional gain andspatial selectivity while employing an array of receiving antennas thatis small as measured in units of the wavelength of radio waves beingreceived or transmitted, especially in the case of spatially oversampled arrays. The inventions can also be generalized to performfrequency/wavenumber, multi-dimensional spectrum analysis, as well asone-dimensional frequency spectrum analysis.

Non-Fourier transformation can be applied in beam formation, or inspectral analysis, as a replacement for conventional Fourier techniques,such as the DFT, to produce superior results under some circumstances.Specifically, a transformation can be selected, or designed, to providea much higher resolution for the array, without the penalties normallyencountered with the super-gain weighting approach. The presentinvention therefore makes use of an alternate form of transform, termedhere a “Hermetic Transform,” in place of a conventional DFT/FFT ortime-domain equivalent, and resolution can be increased, in some casessubstantially, as a result.

Other features and advantages will become apparent from the followingdetailed description and drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic of a conventional method for producing a set ofbeams using complex weights.

FIG. 2 is a schematic relating to the reception of signals with anantenna array.

FIG. 3 is a graph showing an amplitude response from a five-elementarray according to one embodiment.

FIG. 4 is a schematic of an alternative implementation on the receivingside.

FIGS. 5-10 are graphs demonstrating features of embodiments describedtherein.

FIGS. 11-13 are block diagrams of interpolators.

DESCRIPTION

While the process of beam formation can be accomplished using eitheranalog or digital hardware, in modern systems it is often the case thatthe phase-shifting, or equivalent time delay operations, as well asamplitude weighting of signal channels are accomplished usingspecialized digital hardware or programmable processors which processantenna signals that have been filtered, sampled, and digitized. Withoutloss of generality, an embodiment that is completely digital willtherefore be described here to illustrate a representative case.

FIG. 1 indicates a conventional method for producing a set of receivingbeams (Beam 1 and Beam 2) using a combination of complex weights(W_(ij)) that are applied to the digitized beam data. The weights areassumed to be complex, as provided for in the diagram by the ‘QuadratureSampling’ function. These weighted channels are summed to form a set ofspatial filter outputs known as “beams.” The complex weight factorsaccomplish both phase shifting and amplitude weighting of the antennadata prior to summation.

Beamforming architectures normally encountered would typically resemblethat shown in FIG. 1. Sensor elements 10 (e.g., antenna elements in thecase of radar or RF communications systems) are simultaneously sampledand converted to digital form. This function is referred to asQuadrature Sampling 12. The digital form can be put into the form ofIn-Phase and Quadrature (I&Q) data 14, or equivalently complex sampleswhich represent the analytic signal. Next, each channel is multiplyweighted by weights (W_(ij)) and summed in summers 18 to form multiplebeams, essentially accomplishing the matrix multiplications outline inthe discussion above. In the diagram, data is multiplied by one set ofweights and summed to form Beam 1, and multiplied by another set ofweights and summed to form Beam 2. The numbers of sets ofweighting/summation combinations is equal to the number of beams (lookdirections) to be formed.

The weights are complex and each complex weight multiplication can bedecomposed into a real (amplitude) multiplication and a phase shift. Inthis form, the beamforming architecture can be made into an equivalentanalog form, wherein the Quadrature Sampling can be replaced with acomplex mixing (translation) plus band pass filtering, and the phaseshifts could be accomplished using analog phase shifters.

In the application domain of beam formation, the Discrete HermeticTransform (DHT) procedure described here is flexible enough toaccommodate requirements for broadband operation, and/or compensationfor the disruption of wave front shape due to local multipath near thereceiving or transmitting array. The technique can also be made to workfor essentially any array geometry, as shown below. Moreover, the finalelemental beamforming operations of weighting and phase-shifting can beperformed via either analog or digital means, in order to accomplish therequired mathematical effects in transforming channel signal data intospatially filtered ‘beam’ data.

In the area of spatial beamforming, higher gain arrays are made possiblewhile still allowing a small physical array. It is also the case, withinthe application domain of spectral analysis, the DHT achieves a higherfrequency resolution for the same time duration of signal, allowing forthe capture of non-stationary/dynamic signals, or for the increase insignal to noise ratio in narrowband signal detection. The combinedoperations of spatial filtering/beamforming, and spectral analysis isknown as frequency-wavenumber processing, and the technique can begeneralized to perform this combined type of processing as well.

The above discussion does not preclude other applications where signalphasing or resolution enhancement is important, such as coherent opticalsystems, coherent electron or ion microscopy, or signal storage forcomputers or other storage of digital or analog signal data. Forexample, the devices and methods can be applicable to either transmit orreceive beam formation in the acoustic, radio frequency, or opticalspectrum. Because of the wave/particle duality of quantum physics, thetechnique can also be used in electron or ion microscopy imaging and/orbeam shaping. The application of this technology to the spectrumanalysis and filtering of signals to the complementary Fourier transformof signals is applicable in any of the above domains, as well as to datatransformations in more general applications wherever suchtransformation may be useful. This would include fields as far apart asmedical EEG or EKG analysis, image enhancement, stock market and timeseries forecasting, and general pattern analysis and recognition.

A general diagram for a common embodiment is shown in FIG. 2. Aplurality of N antennas 20 form an array of sensors, the outputs whichare provided to Tuning and quadrature Sampling blocks 22 where thesignals are heterodyned down to an intermediate frequency (IF) orbaseband, and synchronously time-sampled in quadrature to provide Nchannels 24 of complex (in-phase and quadrature component) sampled data.Heterodyning can be performed using a set of phase-locked localoscillators in order to preserve phase differences between channels.

A set of FIFO buffer memories 26 stores K complex samples from each ofthe N channels. With time synchronization preserved, data samples fromeach channel are multiplexed in MUX 28 to form K, N-component, complexsignal vectors (“snapshots”), size 1×N, which are subsequently processedby a processor 30. The processor can include, for example, one or moreprogrammable Central Processing Units (CPUs) or digital logicspecialized hardware processing unit(s), such as a Field ProgrammableGate Array(s) (FPGA). The processor performs a complex matrixmultiplication using a matrix of size N×M to create K beam outputvectors 32 of size 1×M. The special data matrix which allows thecreation of super-resolution properties, described in the body of thissection, is stored in memory 34 for use by the processor. Alternativelythe matrix multiplication can be re-arranged to perform M×N matrixmultiplies on N×1 vectors to produce a set of K M×1 complex beamvectors. The complex matrix multiply is referred to here as a DiscreteHermetic Transform (DHT) in analogue to the traditional Discrete FourierTransform (DFT).

Discrete Hermetic Transform (DHT)

Linear discrete time transforms essentially can be characterized interms of matrix multiplications. A technical description of the DHT asapplied to beamforming for a linearly arranged array of radio wavereceiving elements (e.g., a linearly arranged array of antennas)illustrates an application of the technology described here to oneexemplary domain, that of spatially filtering out a communicationssignal, arriving from a specific direction, in the presence ofinterfering signals arriving from other directions.

Assume elementary array geometry, a linear array with uniform spacing ofd spatial units between the N elements of the array, and consider anacoustic (pressure field) plane wave of the formE(r,t)=A e cos(k·r−ωt+φ)  E1.generated by a remote radio wave source, and impinging on the array,where E is the electric field, A is the wave amplitude, e is thepolarization vector, k is the wave-vector having length |k|=ω/c (c beingthe speed of sound) and pointing in the direction of energy propagation,ω is the radian frequency=2π×the wave frequency in Hz, r is the physicalpoint in space where the field value is measured, and φ is an arbitraryphase parameter of the wave.

The above expression can be re-written in the form:E=Re{A eexp[k·r−ωt+φ]}  E2.where E is the real part of a complex exponential. From this pointforward, the wave is considered to be represented by its complex form,understanding that to get to a real physical value one takes the realpart. Only the component of the field in the plane of polarization isconsidered (circularly polarized waves being represented by a sum ofperpendicular polarized components). The reference phases are set to bezero at an arbitrary physical point (r) and reference all other phasevalues to it.

A set of receiving elements for the array are assumed to be located at aset of points in space, which are assumed, for convenience, to lie alongthe x-axis of an x/y/z three dimensional coordinate. If θ is the anglebetween the propagating wave-vector and the axis of the array thenE(x _(n) ,t)=Aexp[kx _(n) sin(θ)−ωt]  E3.where x_(n) is the x-coordinate of the nth element. Finally, anappropriate proportionally is assumed for the receiver to convert field(E) to voltage signal (V), through induced motion of charges in theconductive antenna. One can therefore write the following expression,using x_(n)=n d, with d being the inter-element spacing, and suppressingthe time dependence, for the signal voltage from each receiving channelas follows:V(n;θ)=exp(j[2πn(d/λ)sin(θ)], n=0,1, . . . [N−1]  E4.with j being the square-root of (−1), λ being the wavelength of anarriving plane wave, and λ=c/f.

The array of values V(n; θ)=V(θ) is referred to as the ‘signal vector,’which is taken to be a column vector of length N elements (an N×1 matrixof complex values) and representing a set of received voltages producedby the N receiving elements in response to a pressure wave with harmonictime dependence arriving from a particular direction indicated by thearrival angle θ.

The conventional DFT type beamformer includes a matrix multiplication ofthe form below:Σ(θ₁,θ₂,θ₃, . . . )U(θ)=β(θ)  E5.where each row of the matrix Σ(θ₁, θ₂, θ₃, . . . ) is constructed as a‘matched filter’ for an arriving plane wave, as the complex conjugatetranspose of a signal vector of unit amplitude assumed to be generatedby a wave arriving from a given direction (θ_(i), i=1, 2, 3 . . . ).β(θ) is the pattern response vector, also an N×1 (complex) vector. Thevector U(θ) is the vector of channel voltages as described above,assumed to be generated by an arriving plane acoustic wave, from adirection of arrival that is, a priori, unknown.

Thus the matrix Σ(θ₁, θ₂, θ₃, . . . ) is of a form indicated by theequation below:Σ(θ₁,θ₂,θ₃, . . . )=∥V(θ₁)V(θ₂)V(θ₃) . . . )∥^(H)  E6.where the superscript H in Equation E6 indicates a Hermetian Transposeoperation, i.e., the complex conjugate transpose of the matrix havingcolumns that are of the form of equation E4.

In the above expression, each matched filter vector (V) is normalizedwith the same amplitude (A) which is arbitrary and nominally set to 1(unity). Since each of the vectors V(θ_(i)) are N elements long, thedimensionality of Σ is M (rows)×N (columns) where M corresponds to thenumber of beams (i.e., the number of “look” or “steering” directions)and N is the number of receiving channels used to form each beam.

For the linear array geometry as described above, the matrix Σ(θ₁, θ₂,θ₃, . . . ) is explicitly given by the following expression:

$\begin{matrix}{{\begin{matrix}{\exp\left\lbrack {{- j}\; 2\;{\pi\left( {d/\lambda} \right)}\;{\sin\left( \theta_{1} \right)}} \right\rbrack} & {\exp\left\lbrack {{- j}\; 2\;{{\pi 2}\left( {d/\lambda} \right)}\;{\sin\left( \theta_{1} \right)}} \right\rbrack} & {\exp\left\lbrack {{- j}\; 2\;{{\pi 3}\left( {d/\lambda} \right)}\;{\sin\left( \theta_{1} \right)}} \right\rbrack} & \ldots \\{\exp\left\lbrack {{- j}\; 2\;{\pi\left( {d/\lambda} \right)}\;{\sin\left( \theta_{2} \right)}} \right\rbrack} & {\exp\left\lbrack {{- j}\; 2\;{{\pi 2}\left( {d/\lambda} \right)}\;{\sin\left( \theta_{2} \right)}} \right\rbrack} & {\exp\left\lbrack {{- j}\; 2\;{{\pi 3}\left( {d/\lambda} \right)}\;{\sin\left( \theta_{2} \right)}} \right\rbrack} & \ldots \\{\exp\left\lbrack {{- j}\; 2\;{\pi\left( {d/\lambda} \right)}\;{\sin\left( \theta_{2} \right)}} \right\rbrack} & {\exp\left\lbrack {{- j}\; 2\;{{\pi 2}\left( {d/\lambda} \right)}\;{\sin\left( \theta_{2} \right)}} \right\rbrack} & {\exp\left\lbrack {{- j}\; 2\;{{\pi 3}\left( {d/\lambda} \right)}\;{\sin\left( \theta_{2} \right)}} \right\rbrack} & \ldots \\\ldots & \ldots & \ldots & \ldots\end{matrix}}.} & {E7}\end{matrix}$

If (d/λ) sin(θ) is recognized as a normalized wave number frequency (Ω)then the above matrix can be re-written as:

$\begin{matrix}{{\begin{matrix}{\exp\left\lbrack {{- j}\;\Omega_{1}} \right\rbrack} & {\exp\left\lbrack {- {j\left( {2\;\Omega_{1}} \right)}} \right\rbrack} & {\exp\left\lbrack {{- j}\;\left( {3\;\Omega_{1}} \right)} \right\rbrack} & \ldots \\{\exp\left\lbrack {{- j}\;\Omega_{2}} \right\rbrack} & {\exp\left\lbrack {- {j\left( {2\;\Omega_{2)}} \right\rbrack}} \right.} & {\exp\left\lbrack {- {j\left( {3\Omega_{2}} \right)}} \right\rbrack} & \ldots \\\; & \; & \; & \; \\\ldots & \ldots & \ldots & \ldots\end{matrix}}.} & {E8}\end{matrix}$and the operations produced by the Matrix Multiplication results inDiscrete Fourier Transform of the input vector V={Vn}. Substituting theexplicit form of the matrix into equation E5,

$\begin{matrix}{{\sum\limits_{n}\;{{\exp\left\lbrack {{- j}\; 2\;\pi\; n\;\Omega_{m}} \right\rbrack}U_{n}}} = {{\beta\left( \theta_{m} \right)}.}} & {E\; 9}\end{matrix}$where the sum runs over the index n, i.e., over the component receiverchannels, and the index m indicates the different wave-numberfrequencies, arranged according to the rows of the matrix, and given bythe following equation:Ω_(m)=2π(d/λ)sin(θ_(m))  E10.

From the above expression, one can see that the expression for the beamresponse β(θ_(m)) is in the form of the Discrete Fourier Transform (DFT)of the vector U, according to the normal conventional definition. See,e.g., Gabel and Roberts, “Signals and Linear Systems”, 2nd Edition, JohnWiley & Sons, New York, 1980 (see esp. Equation 5.181 on page 345 for aconventional definition of the DFT).

The DFT of a discrete sequence or function f(n) is defined here,according to the expression below:

$\begin{matrix}{{F(\Omega)} = {\sum\limits_{n = 0}^{N - 1}\;{{f(n)}{{\exp\left\lbrack {j\;\Omega\; n} \right\rbrack}.}}}} & {E\; 11}\end{matrix}$where F(Ω) is the DFT evaluated at frequency Ω.

The expression of equation E5 can also be interpreted as the projectionof the complex vector U onto a set of complex basis vectors e(θ) whichare the rows of the matrix Σ(θ₁, θ₂, θ₃, . . . ).

As discussed above, it is often the case that the channel data (U_(n))are multiplied by a weight function w(n), to produce beams that have lowside lobe response away from the main lobe/axis of the beam formed. Inthis case equation E11 becomes:

$\begin{matrix}{{F(\Omega)} = {\sum\limits_{n = 0}^{N - 1}\;{{w(n)}*{f(n)}{{\exp\left\lbrack {j\;\Omega_{n}} \right\rbrack}.}}}} & {E12}\end{matrix}$

This operation can be introduced in equation E5 through thepre-multiplication of U with a diagonal matrix W (a matrix with valuesof zero outside the main diagonal), the elements of which are w(n):Σ(θ₁,θ₂,θ₃, . . . )W*U(θ)=β(θ)  E13.

Narrowing of the main lobe in beamforming may also be accomplishedthrough the introduction of special weights, as is previously known. Inthis case, the above form of equation E13 still applies, with w(n) equalto the so-called super gain weights.

The expression of equation E13 can be generalized to the case where thematrix W is a non-diagonal matrix, and in fact can be designed withparticular properties so as to improve beam directivity, especially inthe case of arrays which are sampled at less than half-wavelengthspacings. For the case where M is much greater than N, least squarestechniques can be applied to solve for a matrix W which makes the beamresponse β(θ) as close as possible to a reference vector ρ(θ) in a leastsquares sense.

In one embodiment, the following procedure is applied. First, for eachof the chosen beam directions {θ₁, θ₂, θ₃, . . . }, a reference vectorρ(θ_(m)) (a vector of have dimensions N×1) is chosen to have zeroes inall rows, except for the row corresponding to the one selected beamdirection, θ_(m), in which row is placed the value “1” (unity). Thereference vector ρ(θ_(m)) essentially represents an “ideal” beamresponse, with response only in the desired direction, θ_(m), and zeroresponse elsewhere. The following least-squares problem is then solvedto determine the matrix W:Σ(θ₁,θ₂,θ₃, . . . )W*V(θ_(m))=ρ(θ_(m))  E14.to create a matrix that produces a beam response {β(θ)} that is as closeas possible to the desired beam response ρ(θ_(m)) in a least-squaressense.

The least-squares solution to the above problem is directly solved foryielding:w={Σ _(H)Σ)⁻¹Σ^(H)ρ(θ_(m))H[V(θ_(m))]^(H){[V(θ_(m))[V(θ_(m))]^(H)}⁻¹  E15.where the H superscript indicates the Hermitian or complex conjugatetranspose, and the (−1) superscript indicates the matrix inverse of asquare matrix.

The matrix Σ is an (M×N) matrix, with V and ρ are (N×1) matrices. Thematrix W is an (N×N) matrix. Because the square matrices being invertedmay have less than full rank, for purposes of practical computation, thefull matrix inverse is replaced by the pseudo-inverse (−1 goes to #).W={(Σ^(H)Σ)^(#)Σ^(H)}ρ(θ_(m))[V(θ_(m))]^(H){[V(θ_(m))[V(θ_(m))]^(H)}^(#)  E16.

Next, using matrix W as obtained above, for each of the beam directions,a new beamforming matrix (H) is created to replace Σ by replacing theθ_(m) row of Σ with the row vectorh(θ_(m))=[V(θ_(m))]^(H) W(θ_(m))  E17.where W(θ_(m)) is the solution (W) to equation E16 for the beam steeringdirection θ_(m).

Next, the matrix Σ in equation E5 (the fundamental beamforming equation)is now replaced by the matrix H, termed here the Hermetic Matrix, whereH is given by

$\begin{matrix}{H_{({\theta_{1},\theta_{2},\theta_{3},\mspace{11mu}\ldots})} = {{\sum{*w}} = {{\begin{matrix}{h\left( \theta_{1} \right)} \\{h\left( \theta_{2} \right)} \\{h\left( \theta_{3} \right)} \\\ldots\end{matrix}}.}}} & {E\; 18}\end{matrix}$

The Fourier Basis Functions (rows of the Σ matrix) of equation E5 havenow been replaced with a different set of basis functions which haveuseful properties, i.e., they produce a set of beams which are as closeas possible to a set of reference beams (such as the “ideal” beamsdescribed above), in a least-squares sense. The matrix rows are normallyscaled for numerical reasons, e.g., to produce a peak response of unitywhen a steering vector of unity amplitude is presented.

The new beamforming equation that replaces equation E5 above istherefore the following:H(θ₁,θ₂,θ₃, . . . )U(θ)=β(θ)  E19.

Instead of a beamforming method based on Discrete Fourier Transforms,the present method is based on what is termed here a Discrete HermeticTransform, wherein the unknown channel data is pre-multiplied by theHermetic Matrix (H) to produce the beam response vector β(θ), accordingto equation E19.

Applications that employ the conventional DFT (or FFT or Butler Matrix)types of beamforming (or time-delay/sum equivalents) can be made to usea DHT procedure substantially equivalent to that described above.

Other optimization procedures may be envisioned, for example thebackpropagation algorithm used in neural network processing, as well asstepwise recursive implementations of least-squares, such as the Kalmanfilter. While the above treatment is static, i.e. the desired beam shapeis known and the Hermetic Transform Matrix pre-determined, adaptivereference generation, for example, to accomplish null steeringdynamically, can also be implemented.

Although the generalizations are described in terms of beamforming andarrays, they can apply to other domains, such as the time-frequencyanalysis of signals performed to determine the signal spectrum, or toencode information for transmission or storage in communications orcomputer systems.

Multi-Dimensional Data/Arrays

This procedure can be generalized to the case of multi-dimensionaldata/arrays. First, in the ideal case of plane wave beamforming, thematched filter vector for a given row of the initial beamforming matrix(i.e. the steering vector V^(H)) is set to the following:V[n;k(m)]={exp[−jk·r _(n)]}  E20.where there are N elements, indexed by n, and m beam-steer directionswith incoming wave-vectors k(m).

Data from particular element locations in a planar or volumetric array(e.g. indexed by row and column in the case of a planar, rectangulararray) is mapped through a consistent bi-directional indexing scheme,into a single column data vector with a single component index (n).Similarly, beam steering directions (e.g. in azimuth and elevation) arere-indexed into a single component output vector with a single componentindex (m). The prior (HOP) technique for one-dimensional arrays is thenapplied as described above.

Synthetically Constructed Interpolated Arrays (SCIA's)

Because of the Nyquist Sampling Theorem, a band-limited signal which issampled with intervals no larger than the Nyquist rate, can berepresented by, and recovered from, a set of discrete samples that meetthe Nyquist sampling criteria. See, e.g., Gabel and Roberts, Chapter5.12, pages 327-332. According to the Nyquist Sampling Theorem, theminimum sampling frequency (Nyquist Rate) is no higher than twice thehighest frequency present, and can be as low as twice the bandwidth ofthe signal (or the bandwidth of the signal in the case of a complex,analytic representation). This suggests that the oversampling used toincrease the resolution of beams or frequency bins via DHT may begenerated through up-sampling, i.e., through the interpolation of datawhich is already sampled at a rate that is higher than Nyquist. This isa useful result, as the number of physical channels needed canpotentially be much smaller than the number of data channels needed inthe beam forming matrix in order to produce highly directive beams. Theabove concept is referred to here as that of a Synthetically ConstructedInterpolated Array (SCIA). The SCIA can be constructed and the DFTmethod applied to the data with some minor modifications. Specifically,as applied to SCIA, the exact matched filter steering vector (V)utilized in the beamforming matrix Σ(θ₁, θ₂, θ₃, . . . ) as describedabove, is replaced with the interpolated (matched filter) steeringvector. All other aspects of the HOP remain the same. This approachallows the least-squares procedure to compensate for any interpolationerrors due to numerical interpolation filtering. Interpolation ofcomplex data is done in the polar domain, interpolating magnitude andinterpolating phase of the complex vectors.

Doblinger (cited above) shows that interpolated conventional arrays canproduce lower sidelobes through the creation of “virtual” elements.Similar benefits from SCIA interpolation, in terms of mainlobesharpening, can occur when combined with the Hermetic Transformprocessing.

Beamforming with Multipath

A common problem in radar and communications antenna processing is thecorruption and distortion of the arriving wavefront via localreflections and multipath propagation/scattering. In this case, theabove procedure is modified by replacement of the steering vectors rows(V*(θ_(m))) of the beamforming matrix Σ(θ₁, θ₂, θ₃, . . . ) withcalibration vectors, that are either data actually observed from thearrivals the chosen steer angles (θ₁, θ₂, θ₃, . . . ), or are derivedfrom accurate physical models which include the local scattering.

Hybrid Procedure

In the above formulation, the step outlined in equation E17:h(θ_(m))=[V(θ_(m))]^(H) W(θ_(m))  E21can be modified to add to the matrix W(θ_(m)) of equation E16, adiagonal matrix which contains a conventional array weighting functionf(n), i.e., the components of W are modified as follows:W _(nm) >>W _(nm) +f(n)δ_(nm)  E22where δ_(nm) is the kronecker delta, defined to be unity when n=m, andzero otherwise. This hybrid procedure produces a beam shape which isintermediate between the full procedure and that of the conventionalapproach.

The above generalizations can be summarized as follows:

-   -   (1) the methods can be used with multi-dimensional data, such as        with data from planar or volumetric arrays;    -   (2) the methods can be used on Synthetically Constructed        Interpolated Arrays, obtained by interpolation of data from        physical elements;    -   (3) the methods can be used in the presence of, and specifically        to mitigate the effects of, multipath which corrupts and        distorts the signal wavefronts; and    -   (4) Hybrid DHT/conventional approaches can be formulated and        applied as well.

In addition, use of non-planar wavefront matched filters would allow thesystems and methods here to be used to develop beams focused atparticular points in space.

Broadband Application

There are several ways to apply the above technique to broadbandsignals:

-   -   (1) the signals can be processed by a narrowband transform,        e.g., DFT/FFT, or as seen below by DHT, and frequency dependent        matrices used to process transform ‘bins’;    -   (2) the row vectors of the Beam Forming Matrix H (vector used to        form a beam in a given direction) can be characterized so that        the magnitude and phase shift applied by each row component can        be applied by a digital filter, this mapping being accomplished        using standard digital filter design techniques offline;    -   (3) the procedure outlined above for design of the matrix W can        be modified as follows: (a) the Σ matrix is chosen to correspond        to the middle of a frequency band of interest, (b) the vector V        is replaced with a matrix with column vectors corresponding to V        vectors at a number of selected frequencies in the band of        interest, and (c) the response vector p is replaced with a        matrix, the columns of which is the desired or ideal response at        the selected frequencies. The matrix W developed for each beam        steer direction is then the best single least-squares form for        the selected band of interest.        Decomposable Architecture

The Hermetic Matrix H is formed as the product of a normal beamformingmatrix part and another matrix W, which substitutes for the normalweighting used in conventional phased array beamforming to controlsidelobes. When decomposed in this form, the present system and methodeffectively adds a pre-processing step to a conventional beamformingsystem. The Hermetic Beamformer architecture can thus be separated intoa conventional beamformer, preceded by an antenna preprocessor whichperforms a matrix multiplication of W as specified above, times theindividual antenna element time series data. Often this matrix is can beapproximated well by a real-valued matrix; for example, the matrix belowwas developed using a version of HOP for a 5-element linear array withtotal array length 0.5537λ. The amplitude response created for a beamsteered broadside, after the elements have been pre-processed with the Wmatrix below,

$\quad{\begin{matrix}0.0487 & {- 0.1558} & 0.2150 & {- 0.1491} & 0.0442 \\{- 0.1558} & 0.5070 & {- 0.7085} & 0.4971 & {- 0.1491} \\0.2150 & {- 0.7085} & 1.0000 & {- 0.7085} & 0.2150 \\{- 0.1491} & 0.4971 & {- 0.7085} & 0.5070 & {- 0.1558} \\0.0442 & {- 0.1491} & 0.2150 & {- 0.1558} & 0.0487\end{matrix}}$is shown in FIG. 3.

The beam pattern shown here is approximately 36° wide vs. theapproximately 120° wide pattern expected with conventional array beamforming.

Referring to FIG. 4, a diagram of the separable architecture is showngenerally similar to FIG. 2 but with two stages of processing: CPU 34with an N×N matrix multiply to create N channels 36 of “sharpened” datachannels, and an M×N matrix multiply (equivalent), or conventionalbeamforming network/processing stage 38.

Useful Results and Applications of the Invention

A number of useful results can be obtained by applying the aboveprocedure to some common problems in beamforming and to spectrumanalysis. Other evident and useful applications can also be readily seenand described.

Beamforming with Physically Small Arrays

FIG. 5 shows results of beamforming a 2/10 wavelength (total length)linear array. The results plotted are pattern response vs. angle, indecibels (dB) with the peak of the response normalized to one (unity).

The results of conventional beamforming using DFT beamforming anduniform weighting on a 7-element array (seven physical receive channels)is shown by line 40 at the top. The resulting pattern developed for thesame 7-element linear array using the DHT with Hermetic Matrix derivedfrom equation E16 is indicated by curve 42. A peak of the DHT pattern iscorrectly located at the true beam arrival angle. The ‘+’ pointsindicate by comparison, the result of creating a 7-element SyntheticallyConstructed Interpolated Array (SCIA) from only three physical elementsthat span the 2/10 wavelength array (i.e., equal 1/10 wavelengthspacing). These patterns serve to illustrate the utility of the systemsand methods with regards to the increased resolution properties.

The same plot under discussion shows the 3-dB down point as a black line44, so that pattern main lobe widths can be compared. The patternderived is seen to be on the order of 20° wide whereas the pattern fromthe conventional (DFT) approach is nearly omni-directional. Withconventional beam forming approaches, the array would have to be on theorder of 3-wavelengths to obtain a 20° wide beam, or 15 times as long asthe array being considered here. In addition, the ability to utilize asmall number of physical elements results in a reduced hardwarecomplexity of the array, a reduced number of data acquisition channels,etc.

Similar advantages can be obtained for multi-dimensional arrays andarrays of other geometries than linear/planar.

FIG. 6 shows raw amplitude azimuthal beam pattern data and acorresponding theoretical prediction (model) for conventionalphased-array beamforming of a cylindrical array using 5 elementsarranged on a half-face.

FIG. 7 shows three beam amplitude patterns, corresponding to the samecylindrically shaped array. The outer pattern (DHT-5 Staves) shown asline 50 is the same predicted pattern shown in FIG. 6 using conventionalbeamforming and uniform weights. The next pattern shown with line 52 isobtained by applying the present invention to the beamforming of thesame data from the 5-stave used to form the conventional beam. Thesharpest pattern shown with line 54 is formed with a SCIA, interpolatingthe 5-stave data up to 19 total receiving channels using interpolationthen applying the method described herein to form the beam. The resultsshow a significant increase in beam resolution.

Two Dimensional Arrays

Two dimensional DHT processing can be used to form beams fromtwo-dimensional arrays, or to perform beamforming and spectrum analysissimultaneously (frequency-wavenumber spectral analysis).

A two-dimensional planar array example corresponding to the above lineararray example illustrates the applicability and utility of the presentinvention for the multi-dimensional case. FIG. 8 depicts thetwo-dimensional beam pattern amplitude for the case where a 7×7 arraywith 1/28λ spacing between elements has been beam-formed with 60elevation and 60 azimuthal steering directions over an entire half-planefaced by the array. The pattern response to an arriving plane wave isshown in FIG. 8, with amplitude shown on the z axis and angle shown asangle index (approximately 3 degrees per index step). The beam widthobserved is clearly far superior to that of a conventionally beam-formedarray of the same dimensions, which would be essentially omnidirectionalin its response.

Spectrum Analysis Using DHT

A conventional method of performing spectrum analysis is through theDiscrete Fourier Transform (DFT), usually through a Fast-FourierTransform (FFT) implementation of the DFT. The present methods andsystems can be utilized for frequency spectrum analysis directly throughthe application of the Hermetic Matrix and the Discrete HermeticTransform (DHT). In this case, the discrete-time signal can over-sampledat the Analog-to-Digital Conversion (ADC) stage, or is upsampled viainterpolation filtering.

FIG. 9 illustrates the effect of applying various sample rates to theDHT of an 8 MHz sinusoidal (CW) signal that is 0.1 msec in duration,with results shown terms of the square of the transform values convertedto dB.

In conventional DFT/FFT spectrum analysis, when a signal is T secondslong, the frequency resolution resulting from application of thetransformation is (1/T) Hz in the frequency domain. Thus the expectedfrequency resolution on this signal segment is expected to be on theorder of 10 MHz.

The dashed line 60 in FIG. 9 is the result of applying the DHT to thesignal segment using 2× Nyquist rate. Two darker curves 62 and 64 (thatlargely overlap) are shown representing a 16× Nyquist sample/sec rate,and another curve being intermediate between the above two values. Assampling rate is increased, the resolution eventually converges to aminimum value. The flat line 66 shows the −3 dB down point relative tothe maximum point of the frequency response, and the resolution is onthe order of 1250 Mz wide vs. the expected 10,000 MHz resolution. Interms of signal to noise ratio, when using the DHT for detection, theapproximately 8:1 resolution/bandwidth improvement is expected toproduce a 10 log₁₀(8) or about 9 dB improvement in signal to noiseratio.

FIG. 10 shows a direct comparison of the results obtained via DFT/FFTfor the same example, as compared to the convergent DHT result shownabove. The top curve 70 shows the square of the Discrete FourierTransform Magnitude output (periodogram) converted to dB, while thelower curve 72 with a series of peaks shows square of the DHT magnitudespectrum also in dB, both for same case of a 0.1 msec microsecondduration, 8 MHz sinusoidal (CW) signal. As expected, the resolution ofthe DFT/FFT is approximately 10 MHz.

Noise

Use of the Hermetic Transform as described above allows the creation ofnarrow receiving beams with a multi-element sensor array, where thesensor can include antennas, microphones, geophones, etc. Data iscaptured from a set of elements. In one embodiment and as shown in FIG.2, the data is captured digitally at the output of synchronously sampledAnalog to Digital Converters (ADCs), and converted to quadrature sampled(complex) data on a per channel/sensor basis. The data is then bandpassfiltered (or narrow-banded, e.g., using an FFT) and a Hermetic Transformis applied to the bandpass channel(s) to produce a set of beams,spatially filtering and channelizing incoming signals impinging on thearray according to their direction of arrival (DOA). The HermeticTransform H is in general a complex-valued matrix of dimension M (numberof beams)×N (number of sampled sensor elements) and has the general formH=Σ^(H)W  E23where the Σ matrix has a set of columns, each column being made up of aset of complex phasors which are in effect a spatial matched filter tothe an incoming signal from a specified beam direction, Σ^(H) is theHermetian Conjugate (complex conjugate transpose) of the Σ matrix, and Wis in general a full-rank (N×N) complex valued matrix which performsbeam sharpening to fulfill the minimization of an objective functionwhich is preset in the design of the transform. In one embodiment, thesolution for W is found by solving in a least-squares sense, thefollowing equation:Σ^(H)WΣ=CI  E24where C is a scaling constant and I is the identity matrix (M×M). Thetransformation of a snapshot (time-sampled data from N elements) intobeams is then just accomplished using the Hermetic Transform in the formof a matrix multiplication, i.e.B=H s  E25where B is the a set of vector of time series from the M beams, H is theHermetic Transform matrix as indicated above, and s is a vector of timeseries from each of the sensor elements.

It is found empirically that the above approach in finding W can beunsatisfactory with respect to its sensitivity to noise in the sensordata, as it can be with known Super-Gain or Super-Directive beam-formedarrays.

This effect can be mitigated by placing an additional matrix between thedata vector s and the Hermetic Transform H in order to better conditionthe data and ensure good performance of the beamformer in dealing withnoisy signals.

The new equation replacing the algorithm of equation E25 has thefollowing form:B=H N s  E26with the matrix HN being referred to as a noise-compensated HermeticTransform. In one embodiment, a solution for N is found by firstgenerating a matrix Z of the same dimension as the matrix Σ, butincluding noise of a particular character, for example with elementsthat are complex white Gaussian noise distribution; the matrix Z can beused to solve for N in a least-squares sense with the equation shownbelow:N(Z+Σ)=ΣThe form of this equation can be made more general, with the expression,N[(Z ₁+Σ)(Z ₂+Σ) . . . ]=[ΣΣ . . . ]with the various Z_(i) being different noise realizations generatednumerically using a pseudo-random number generator. The equals sign inequations (4) and (5) does not mean to indicate actual equality, butinstead indicates the equation to be solved in a least-squares sense.Interpolation

In the hermetic transform approach to beam formation of spectrumanalysis, the signal is over-sampled, with elements less than 1/λspacing in the case of beam forming, or with sampling rates much fasterthan the Nyquist rate in the case of time or frequency domainprocessing. It is often a real-world constraint that the number ofphysical antenna elements and associated ADC channels (in the case ofbeam forming) or the sampling rate (in the case of time and frequencyprocessing) are limited.

Therefore an interpolation process can be added to the HermeticTransform processing system to create signals synthetically which“fill-in” missing data using a smaller number of elements or samples.

FIG. 11 indicates the general structure of a spatially interpolatingbeamformer.

First, data from N antenna elements is buffered into N-element vectorsof complex (I&Q) data. Next, N×N complex-valued matrices which comprisespatial filters for a set of directional sectors (on the order of 5sectors to cover 180°) are applied to the data to filter out onlyarrivals from each sector. The filters can utilize N-element HermeticTransforms. Next, optimal interpolation matrices are applied to producethe signal which is as close as possible according to a particularselected objective (cost) function, to the signal which would have beencaptured at m elements (m>N) which span the same physical space occupiedby the N element array. Typically, a Least-Squares type cost function isutilized in practice. Each interpolation function is itself constructedusing an equation of the typeM S_(o)=S_(i)where the matrix M is the (m×N dimensioned) interpolation matrix to besolved for, S_(o) is has a set of N×1 column vectors corresponding toelement data at the original sampling rate from specific arrival angleswithin the sector in question, post-filtering, and S_(i) is thetheoretically derived set of m×1 column vectors corresponding to theideal element data at the new (higher sampling rate) spacing. For theleast-squares objective function (least-squares minimization) a solutionfor M is given by the following equation:M=S _(o) ^(H) [S _(i) S _(o) ^(H)]^(#)where the superscript H indicates the complex conjugate transpose(Hermitian conjugate) operation and the # symbol indicates thepseudo-inverse operation.

In an actual interpolator, the pre-stored matrixes to perform sectorfiltering are applied, followed by the interpolating matrices; these canbe combined into a single matrix for each sector. The output M×1 datavectors from all sectors can then be vector added (summed together bycomponents) and the result provided to a Hermetic Transform to providethe beamforming.

The features of filtering spatially, followed by interpolatingspatially, can be broken into steps which can produce superior resultsto a single step process. For example, in going from a 3-element antennaarray to an array which is synthetically generated to have theequivalent of 31 elements, one could be prudent to break this intostages, for example spatially filtering into 3-5 sectors, interpolatingeach of these into 13 elements, summing, filtering the output of thisarray into 6-10 sectors, and interpolating the subsequent filtered“arrays” up to 31 elements and summing to combine the 6-10 independentresults. In practical terms all of the summing and multiplication ofmatrices can be normally combined into one single interpolation matrixbut the design of this one matrix involves the design of a set ofstages.

FIG. 12 indicates an exemplary architecture of this approach with acascade of filter/interpolation matrices. Results can be superiorbecause resolution of the beam formation process improves at each stage,and subsequent stages of filtering result in improved separation ofspatial regions, so that optimal interpolators can be applied innarrower sectors, thereby reducing the interpolation error power.

Interferometric Hermetic Beam Forming and Direction Finding

A beamforming device and method is described based on a HermeticTransform derived from calibration data including signal arrivals atparticular frequencies, rather than calculated/modeled signal arrivals.The array can have arbitrary geometry. The beams can be used toaccomplish direction finding. The signals can work with any types ofwaves, for example, radio frequency or acoustic.

As described above, the Hermetic Transform H is in general acomplex-valued matrix of dimension M (number of beams)×N (number ofsampled sensor elements) and has the general formH=Σ^(H) Wwhere the Σ matrix has a set of columns, each column being made up of aset of complex phasors which are in effect a spatial matched filter tothe an incoming signal from a specified beam direction, Σ^(H) is theHermitian Conjugate (complex conjugate transpose) of the Σ matrix, and Wis in general a full-rank (N×N) complex valued matrix which performsbeam sharpening to fulfill the minimization of an objective functionwhich is preset in the design of the transform. In one embodiment, thesolution for W is found by solving in a least-squares sense, thefollowing equation:Σ^(H)WΣ=CIwhere C is a scaling constant and I is the identity matrix (M×M). Theequation for I isI _(k m)=δ(k,m)where δ is the Kronecker delta function, essentially corresponding to anideal beam response for each reference; i.e., the goal for a particularbeam is to give a unit response in a particular reference direction andzero otherwise. In the general case the identity matrix can be replacedwith whatever reference response is desired, such functions thatapproach a delta function. Zeroes can be placed in these references tocompel nulling in particular directions, as another example of aparticular embodiment.

The transformation of a snapshot (time-sampled data from N elements)into beams can be accomplished using the Hermetic Transform in the formof a Matrix Multiplication, i.e.B=H swhere B is the a set of vector of time series from the M beams, H is theHermetic Transform matrix as indicated above, and s is a vector of timeseries from each of the sensor elements.

In the above treatment, the signal reference matrix Σ can be derivedfrom mathematical analysis, simulation, or modeling of the array inquestion. In one embodiment, a signal reference matrix is obtained byactual measurement of signal arrivals at the sensors (antenna elements)at a set of frequencies for a set of directions around 360° azimuth andwhere elevation angles. The above equations still hold. The performancein beam reduction can be greatly improved when an antenna is placed in acomplex scattering environment, for example on an airplane fuselage orin a cell base station location in an urban area, because of thecomplexity of the basis functions where signal arrivals having a greatdeal of multipath interference. In empirical data collections, anadditional three to five fold beam width reduction has been observed.FIG. 13 below shows an actual beam constructed from a set of measuredarrivals using a 4-element antenna. The array size is less than 1/3wavelength in diameter.

The multipath actually helps improve in this case and the beamformingapproach that results is actually based on a form of interferometry.

One additional feature of this approach, termed here “InterferometricHermetic Beamforming,” is that the beams generated are sufficientlynarrow that they can be used for Direction Finding (DF). Either the peakresponse or the first moment of the beam power response (the lattercorresponding to the wavenumber power spectral density) can be used insingle-emitter DF. The existence of multiple peaks would indicatemultiple emitters that could be independently listened by selectingbeams corresponding to peak locations.

With respect to one useful application of this direction findingcapability, one can consider a cellular communications scenario, whereina full set of DHT beams can be formed for a cell tower to direction findand locate a handset emitter and/or the handset emitter to locate thecell tower. This allows the handset to generate and form only the beamnecessary to listen to the cell tower thus greatly reducing real-timecomplexity.

In most cases it would be logical for the cell tower to direction findon the handset uplink channel and pass the relative bearing informationto the cellular handset to use in selecting a beam to listen to. If thecellular handset can access the Global Positioning System (GPS), thenpassing the GPS coordinates of the controlling base-station antenna tothe handset would be sufficient for the handset to adjust beam selectionuntil handoff to a new cell tower. It is often the case that multiplecell towers will operate in combination during transitions and therelative line of bearings from the handset to each cell tower and viceversa, could be used to actually improve on the handset's GPS estimateof its location via DF Fixing/Emitter Location algorithms that aregenerally known.

All of the above discussion works for both radio frequency and acousticforms of waves. In the case of air acoustics, a particular applicationof interest is the location of gunshot noises in order to protectsoldiers from sniper attacks.

Single RF Channel Implementation

The systems and methods described above can be used with a single RFchannel implementation, such as that described in “a compact digitalbeam forming SMILE array for mobile communications,” Goshi, et al, IEEETransactions on Microwave Theory and Techniques, Vol. 52, No. 12,December 2004. As described in the article, which is incorporated hereinby reference, an N-element antenna array can be used with a multiplexingfeed network, a digital sequence generator, and a single RF channel. Thesingle RF channel includes a low noise amplifier (LNA) and a mixer fordown conversion. In an IF-baseband processing block, the signals aredemultiplexed, provided to low pass filters (LPFs) to recover originalsignals, converted into digital form in ADCs, and then provided to adigital beam forming block. These components would generally replace thetuning and quadrature sampling blocks 22 shown in FIG. 2. As describedin the article, to avoid aliasing, each signal is switched at or abovethe Nyquist rate with respect to the modulation bandwidth such that thefrequency is greater than the product of N channels switched in eachcycle and single bandwidth B. This implementation is particularlydesirable to produce needed hardware and can be especially useful wheresmaller size is an improvement, such as in a handset or other compactdevice.

Cellular Communication Implementation

While there have been a number of references to the use of thetechnologies described above for implementations with radiocommunications and cellular communications, this section summarizes someof the features and benefits in this field.

As is generally known, cell phones are interference sources to othercell phones, and other RF interference competes with cell phone signalsas well. A cellular antenna collects all signals in its field of view,which can be omni-directional(360°) or based on some numbers of sectors,often three sectors of 120° each. With the systems and methods describedabove, narrow beam formation can be added to current technology to spanthe station or sectors with a set of narrow spatial beams, with a beamonly sensing the signal in a narrow field of view. In this case, thefield of view, rather than being 360° or 120°, now becomes a beam width.It is desirable for this beam width to be about 10° or less, but it canalso be about 5° or less or even 3° or less.

By using azimuthal beam formation for spatial discrimination, a signalto noise ratio can be increased for better quality and spectralefficiency. The signal to noise ratio gain can be greater than about 5DB or greater than about 10 DB, or up to about 20 DB. The array gainagainst RF interference is proportional to the reduction in the beamwidth. By using geo-location technology such as GPS, the beam formingcan be coordinated between the handset and the base station.

Currently, a cellular base station forms multiple simultaneous beams tospan the field of view, e.g., a 120° sector. If the cell phone can belocated by the base station, the base station can direct the handsetwhere to look with a single beam. Compared to other beam formingapproaches, the methods described above allow an entire group ofrequired beams to performed in a single step with one linear transform.The systems and methods can result in higher resolution, i.e., reducedbeam width, for a constrained antenna size and can produce greater arraygain. The system can form multiple beams or a single beam at a cellularhandset unit.

The systems and methods described above are also useful with multipathenvironments, such as cellular communications. As is generally known,with multipath propagation, signals can combine in a destructive mannerthereby creating fading. These multiple paths can arise from variousreflections. For example, there could be a clear line of sight (LOS)between a handset and a cellular tower, but the signal from the celltower, in addition to having a direct path to a handset, could alsoreflect off of buildings or other structures and thus be received at aslightly different time from the direct path. The use of retransmission,error correcting codes, rake receivers, and methods are all known fordealing with multipath. The systems and methods described above can helpto mitigate multipath fading, and coordination between a base stationand a handset can allow a single beam to be formed at the handset. Asnoted above, the systems and methods can use a small number of elementsand can interpolate spatially to give a benefit that is similar to theuse of a much larger number of physical elements. In one example, threeantenna elements can be used. After I and Q sampling is done, there aretime samples from the three antenna elements. These elements can beprovided to an interpolation filter that can increase the number, forexample, by a factor of 2 or more, a factor of 3 or more, a factor of 4or more, 5 or more, or 6 or more. In one example, the interpolationfilter is used to create a “synthetic array” of 19 elements. The DHTcomplex matrix multipliable block can further increase the effectivenumber of beams, thereby producing I and Q samples for an even greaternumber of beams, such as 91 beams.

The inventions described above can be implemented in one or more of thefollowing: general purpose processors, application-specific integratedcircuits (ASICs), data cards, controllers, processors, and any otherprocessing device, referred to here generally as a processor. Thefunctions can be implemented in any necessary combination of hardwareand software. To the extent that the inventions are implemented in wholeor in part with software that can be used on different types of generalpurpose processors, the software can be kept on medium, such as magneticdisc, optical disc, or solid state memory, and implemented on a computersuch that when installed the software is executed as a series of stepson such processor. Various processes, such as analog to digitalconverters and sampling functionality can occur within dedicatedhardware processes, and/or can be implemented in a form of programprocessor, any of which would constitute circuitry for performing thefunction.

While certain embodiments have been described, others are within thescope of the claims.

The invention claimed is:
 1. A system for coupling to an array of Nantennas for receiving N signals, where N is a natural number greaterthan or equal to three, for receiving signals from a transmittingdevice, the system comprising: circuitry for deriving digital data fromthe N antennas; a spatial filter derived from one or more HermeticTransforms for obtaining signals from the circuitry; a spatialinterpolator comprised of a matrix of complex values for receivingsignals derived from the digital data and providing M interpolatedsignals, where M>N, M representing a set of beams; a selector forselecting one of the beams to identify a location of the transmittingdevice; and a second spatial filter for receiving output signals derivedfrom the spatial interpolator, and a second spatial interpolator forreceiving the filtered signals from the second spatial filter andproviding P beams, where P>M.
 2. The system of claim 1, wherein thesystem is adapted to receive radiofrequency (RF) signals.
 3. A methodfor receiving signals from a transmitting device, the receiving for usewith an array of N antennas for receiving N signals, where N is anatural number greater than or equal to three, the method comprising:sampling the signals to derive digital data from the N elements; spatialfiltering with a spatial filter derived from one or more HermeticTransforms for filtering sampled signals; spatially interpolating with aspatial interpolator comprised of a matrix of complex values the Nsignals to provide M interpolated signals, where M>N, M representing aset of beams; and selecting one of the beams to identify a location ofthe transmitting device.
 4. The method of claim 3, wherein theinterpolating includes applying a matrix based on a selected costfunction.
 5. The method of claim 3, wherein the sampling includesspatial oversampling.
 6. A method for receiving signals from atransmitting device, the receiving operable with an array of N antennasproviding N signals, where N is a natural number greater than or equalto three, the method for receiving comprising: sampling the signals toderive digital data from the N elements; spatial filtering with aspatial filter derived from one or more Hermetic Transforms forobtaining signals from a desired sector; spatially interpolating with aspatial interpolator comprised of a matrix of complex values the Nsignals to provide M interpolated signals, where M>N, M representing aset of beams; selecting one of the beams to identify a location of thetransmitting device; and a second spatial filtering for receiving outputsignals derived from the spatial interpolating, and a second spatialinterpolating for receiving the filtered signals from the second spatialfiltering and providing P beams, where P>M.
 7. The method of claim 5,wherein the interpolating includes applying a matrix based on a selectedcost function.